The drawing shows the top view of two doors. The doors are uniform and identical

Question

The drawing shows the top view of two doors. The doors are uniform and identical. Door A rotates about an axis through its left edge, and door B rotates about an axis through its center. The same force is applied perpendicular to each door at its right edge, and the force remains perpendicular as the door turns. No other force affects the rotation of either door. Starting from rest, door A rotates through a certain angle in 3.60 s. How long does it take door B (also starting from rest) to rotate through the same angle?

Explanation / Answer

the equations of motion are the same for either door. The difference will be the moment of inertia of the door based on the point of rotation, and the torque distance from the axis of rotation will be different

either way use
?=.5*?*t^2
and
F*d=I*?
or
t^2=2*?*I/(F*d)

for door A
3.6^2=2*?*IA/(F*L)
for door B
t^2=4*?*IB/(F*L)
divide
t^2/3.6^2=2*IB/IA

IA=m*L^2/3
and
IB=m*L^2/12

IB/IA=1/4
t^2/3.6^2=1/2

t=3.6/sqrt(2) =2.545 sec

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